We always use the sheets we have with no issues (usually fitted) or you could get sheets that are not fitted. Not sure what you mean by nothing to wrap the sheets around.
But if the sofa mattress is not adequate you can get up to a 4" foam topper.
The DW learned how to make beds in a hospital (nursing school in the 1970s) and on the farm and can do it easily.
We have had about 6 campers over 40 years do not sweat about the little things.